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Baron Baron is a rank of nobility Nobility is a social class normally ranked immediately below Royal family, royalty and found in some societies that have a formal aristocracy (class), aristocracy. Nobility has often been an Estates of the ...

Baron
Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
, engineer, and
physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at leas ...

physicist
who made pioneering contributions to several branches of mathematics, including
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
and
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis Cauchy was the first to for ...
. He was one of the first to state and rigorously prove theorems of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, rejecting the heuristic principle of the
generality of algebraIn the history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of ...
of earlier authors. He almost singlehandedly founded
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
and the study of
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
. A profound mathematician, Cauchy had a great influence over his contemporaries and successors;
Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German The history of the Jews in Germany goes back at least to the year 321, and continued through the Early Middle Ages (5th to 10th centuries CE) and High Middle Ages (''c ...

Hans Freudenthal
stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in
elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the clu ...
alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
.


Biography


Youth and education

Cauchy was the son of
Louis François Cauchy Louis François Cauchy (27 May 1760 – 28 December 1848) was a senior French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, R ...
(1760–1848) and Marie-Madeleine Desestre. Cauchy had two brothers: Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847 and a judge of the court of cassation in 1849, and Eugene François Cauchy (1802–1877), a publicist who also wrote several mathematical works. Cauchy married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. They had two daughters, Marie Françoise Alicia (1819) and Marie Mathilde (1823). Cauchy's father was a high official in the Parisian Police of the
Ancien Régime The Ancien Régime (; ; literally "old rule"), also known as the Old Regime was the political Politics (from , ) is the set of activities that are associated with Decision-making, making decisions in Social group, groups, or other forms o ...
, but lost this position due to the
French Revolution The French Revolution ( ) was a period of radical political and societal change in France France (), officially the French Republic (french: link=no, République française), is a spanning and in the and the , and s. Its ...

French Revolution
(July 14, 1789), which broke out one month before Augustin-Louis was born. The Cauchy family survived the revolution and the following
Reign of Terror The Reign of Terror, commonly called The Terror (french: link=no, la Terreur), was a period of the French Revolution The French Revolution ( ) was a period of radical political and societal change in France France (), offici ...
(1793–94) by escaping to
Arcueil Arcueil () is a Communes of France, commune in the Val-de-Marne Departments of France, department in the southern suburbs of Paris, France. It is located from the Kilometre Zero, center of Paris. Name The name Arcueil was recorded for the fi ...
, where Cauchy received his first education, from his father. After the execution of
Robespierre Maximilien François Marie Isidore de Robespierre (; 6 May 1758 – 28 July 1794) was a French lawyer A lawyer or attorney is a person who practices law, as an advocate, attorney at lawAttorney at law or attorney-at-law, usually ab ...

Robespierre
(1794), it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job in 1800, and quickly moved up the ranks. When
Napoleon Bonaparte Napoleon Bonaparte ; co, Napulione Buonaparte. (born Napoleone di Buonaparte; 15 August 1769 – 5 May 1821) was a French military and political leader who rose to prominence during the French Revolution The French Revolution ( ) r ...

Napoleon Bonaparte
came to power (1799), Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ) is a classical language belonging to the I ...

Laplace
(who is now better known for his work on mathematical physics). The famous mathematician
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

Lagrange
was also a friend of the Cauchy family. On Lagrange's advice, Augustin-Louis was enrolled in the
École Centrale du Panthéon École may refer to: * an elementary school in the French educational stages Educational stages are subdivisions of formal learning, typically covering early childhood education, primary education, secondary education and tertiary education. Th ...
, the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages; the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and the humanities. In spite of these successes, Augustin-Louis chose an engineering career, and prepared himself for the entrance examination to the
École Polytechnique The École Polytechnique (French: l'École polytechnique, commonly known as la Polytechnique or l'X ) is one of the most prestigious and selective grandes écoles Grandes may refer to: *Agustín Muñoz Grandes, Spanish general and politician ...
. In 1805, he placed second of 293 applicants on this exam and was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. Nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées (School for Bridges and Roads). He graduated in civil engineering, with the highest honors.


Engineering days

After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Augustin-Louis stayed for three years, and was assigned the
Ourcq Canal
Ourcq Canal
project and the
Saint-Cloud Bridge
Saint-Cloud Bridge
project, and worked at the Harbor of Cherbourg. Although he had an extremely busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the ''Première Classe'' (First Class) of the
Institut de France The (; Institute of France) is a French learned society, grouping five , including the Académie Française. It was established in 1795 at the direction of the National Convention. Located on the Quai de Conti in the 6th arrondissement of Par ...

Institut de France
. Cauchy's first two manuscripts (on
polyhedra In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
) were accepted; the third one (on directrices of
conic sections In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, the par ...

conic sections
) was rejected. In September 1812, now 23 years old, Cauchy returned to Paris after becoming ill from overwork. Another reason for his return to the capital was that he was losing his interest in his engineering job, being more and more attracted to the abstract beauty of mathematics; in Paris, he would have a much better chance to find a mathematics related position. Therefore, when his health improved in 1813, Cauchy chose to not return to Cherbourg. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior. The next three years Augustin-Louis was mainly on unpaid sick leave, and spent his time quite fruitfully, working on mathematics (on the related topics of symmetric functions, the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, and the newly installed Bourbon king
Louis XVIII Louis XVIII (Louis Stanislas Xavier; 17 November 1755 – 16 September 1824), known as the Desired (), was King of France The monarchs of the Kingdom of France The Kingdom of France ( fro, Reaume de France, frm, Royaulme de Franc ...
took the restoration in hand. The
Académie des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discip ...
was re-established in March 1816;
Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...

Lazare Carnot
and
Gaspard Monge Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing Technical drawing, drafting or drawing, i ...

Gaspard Monge
were removed from this Academy for political reasons, and the king appointed Cauchy to take the place of one of them. The reaction of Cauchy's peers was harsh; they considered the acceptance of his membership in the Academy an outrage, and Cauchy thereby created many enemies in scientific circles.


Professor at École Polytechnique

In November 1815,
Louis Poinsot 250px, ''Louis Poinsot. Lithograph.'' Louis Poinsot (3 January 1777 – 5 December 1859) was a France, French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body cou ...
, who was an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by then a rising mathematical star, who certainly merited a professorship. One of his great successes at that time was the proof of
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

Fermat
's polygonal number theorem. However, the fact that Cauchy was known to be very loyal to the Bourbons doubtless also helped him in becoming the successor of Poinsot. He finally quit his engineering job, and received a one-year contract for teaching mathematics to second-year students of the École Polytechnique. In 1816, this Bonapartist, non-religious school was reorganized, and several liberal professors were fired; the reactionary Cauchy was promoted to full professor. When Cauchy was 28 years old, he was still living with his parents. His father found it high time for his son to marry; he found him a suitable bride, Aloïse de Bure, five years his junior. The de Bure family were printers and booksellers, and published most of Cauchy's works. Aloïse and Augustin were married on April 4, 1818, with great Roman Catholic pomp and ceremony, in the Church of Saint-Sulpice. In 1819 the couple's first daughter, Marie Françoise Alicia, was born, and in 1823 the second and last daughter, Marie Mathilde. The conservative political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died, and was succeeded by his even more reactionary brother
Charles X Charles X (born Charles Philippe, Count of Artois; 9 October 1757 – 6November 1836) was King of France The monarchs of the Kingdom of France The Kingdom of France ( fro, Reaume de France, frm, Royaulme de France, french: link=no, Ro ...

Charles X
. During these years Cauchy was highly productive, and published one important mathematical treatise after another. He received cross-appointments at the
Collège de France The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment (''grand établissement'') in France. It is located in Paris, i ...
, and the .


In exile

In July 1830, the
July Revolution The French Revolution of 1830, also known as the July Revolution (), Second French Revolution or in French ("Three Glorious ays), was a second French Revolution after the First First or 1st is the ordinal form of the number one (#1). Fir ...
occurred in France. Charles X fled the country, and was succeeded by the non-Bourbon king
Louis-Philippe Louis Philippe I (6 October 1773 – 26 August 1850) was King of the French from 1830 to 1848, the last King and penultimate monarch of France. As Duke of Chartres he distinguished himself commanding troops during the Revolutionary Wars, b ...

Louis-Philippe
(of the
House of Orléans The 4th House of Orléans (french: Maison d'Orléans), sometimes called the House of Bourbon-Orléans (french: link=no, Maison de Bourbon-Orléans) to distinguish it, is the fourth holder of a surname previously used by several branches of the R ...
). Riots, in which uniformed students of the École Polytechnique took an active part, raged close to Cauchy's home in Paris. These events marked a turning point in Cauchy's life, and a break in his mathematical productivity. Cauchy, shaken by the fall of the government, and moved by a deep hatred of the liberals who were taking power, left Paris to go abroad, leaving his family behind. He spent a short time at
Fribourg , Location of , Location of () (; frp, Fribôrg or ) or (also called ''Freiburg im Üechtland'' ( gsw, label=Swiss German, Frybùrg ; it, Friburgo or ; rm, Friburg) (not to be confused with Freiburg im Breisgau) is the capital of the Cant ...

Fribourg
in Switzerland, where he had to decide whether he would swear a required oath of allegiance to the new regime. He refused to do this, and consequently lost all his positions in Paris, except his membership of the Academy, for which an oath was not required. In 1831 Cauchy went to the Italian city of Turin, and after some time there, he accepted an offer from the
King of Sardinia The following is a list of rulers of Sardinia Sardinia ( ; it, Sardegna ; sc, Sardigna or ) is the Mediterranean islands#By area, second-largest island in the Mediterranean Sea, after Sicily, and one of the Regions of Italy, 20 regions ...
(who ruled Turin and the surrounding Piedmont region) for a chair of theoretical physics, which was created especially for him. He taught in Turin during 1832–1833. In 1831, he was elected a foreign member of the
Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( Swedish: ''Kungliga Vetenskapsakademien'') is one of the royal academies of Sweden Sweden ( sv, Sverige ), officially the Kingdom of Sweden ( sv, links=no, Konungariket Sverige ), is a Nordic co ...
, and the following year a Foreign Honorary Member of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences, founded 1780, (abbreviation: AAAS) is one of the oldest learned societies A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization ...

American Academy of Arts and Sciences
. In August 1833 Cauchy left Turin for
Prague Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and largest city A city is a large human settlement In geography, statistics and archaeology, a settlement, locality or populated place is a community in which people ...

Prague
, to become the science tutor of the thirteen-year-old Duke of Bordeaux Henri d'Artois (1820–1883), the exiled Crown Prince and grandson of Charles X. As a professor of the École Polytechnique, Cauchy had been a notoriously bad lecturer, assuming levels of understanding that only a few of his best students could reach, and cramming his allotted time with too much material. The young Duke had neither taste nor talent for either mathematics or science, so student and teacher were a perfect mismatch. Although Cauchy took his mission very seriously, he did this with great clumsiness, and with surprising lack of authority over the Duke. During his civil engineering days, Cauchy once had been briefly in charge of repairing a few of the Parisian sewers, and he made the mistake of mentioning this to his pupil; with great malice, the young Duke went about saying Mister Cauchy started his career in the sewers of Paris. His role as tutor lasted until the Duke became eighteen years old, in September 1838. Cauchy did hardly any research during those five years, while the Duke acquired a lifelong dislike of mathematics. The only good that came out of this episode was Cauchy's promotion to
baron Baron is a rank of nobility Nobility is a social class normally ranked immediately below Royal family, royalty and found in some societies that have a formal aristocracy (class), aristocracy. Nobility has often been an Estates of th ...

baron
, a title by which Cauchy set great store. In 1834, his wife and two daughters moved to Prague, and Cauchy was finally reunited with his family after four years in exile.


Last years

Cauchy returned to Paris and his position at the Academy of Sciences late in 1838. He could not regain his teaching positions, because he still refused to swear an oath of allegiance. In August 1839 a vacancy appeared in the
Bureau des Longitudes The ''Bureau des Longitudes'' () is a France, French scientific institution, founded by decree of 25 June 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. During th ...
. This Bureau bore some resemblance to the Academy; for instance, it had the right to co-opt its members. Further, it was believed that members of the Bureau could "forget about" the oath of allegiance, although formally, unlike the Academicians, they were obliged to take it. The Bureau des Longitudes was an organization founded in 1795 to solve the problem of determining position at sea — mainly the
longitudinal Longitudinal is a geometric term of location which may refer to: * Longitude ** Line of longitude, also called a meridian (geography), meridian * Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long ...

longitudinal
coordinate, since
latitude In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the ...

latitude
is easily determined from the position of the sun. Since it was thought that position at sea was best determined by astronomical observations, the Bureau had developed into an organization resembling an academy of astronomical sciences. In November 1839 Cauchy was elected to the Bureau, and discovered immediately that the matter of the oath was not so easily dispensed with. Without his oath, the king refused to approve his election. For four years Cauchy was in the position of being elected but not approved; accordingly, he was not a formal member of the Bureau, did not receive payment, could not participate in meetings, and could not submit papers. Still Cauchy refused to take any oaths; however, he did feel loyal enough to direct his research to
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motion (physics), motions of celestial object, objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical obje ...
. In 1840, he presented a dozen papers on this topic to the Academy. He also described and illustrated the
signed-digit representation In mathematical notation Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, ana ...
of numbers, an innovation presented in England in 1727 by
John Colson John Colson (1680 – 20 January 1760) was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, ...
. The confounded membership of the Bureau lasted until the end of 1843, when Cauchy was finally replaced by Poinsot. Throughout the nineteenth century the French educational system struggled over the separation of church and state. After losing control of the public education system, the Catholic Church sought to establish its own branch of education and found in Cauchy a staunch and illustrious ally. He lent his prestige and knowledge to the École Normale Écclésiastique, a school in Paris run by Jesuits, for training teachers for their colleges. He also took part in the founding of the
Institut Catholique
Institut Catholique
. The purpose of this institute was to counter the effects of the absence of Catholic university education in France. These activities did not make Cauchy popular with his colleagues, who, on the whole, supported
the Enlightenment The Age of Enlightenment (also known as the Age of Reason or simply the Enlightenment); ger, Aufklärung, "Enlightenment"; it, L'Illuminismo, "Enlightenment"; pl, Oświecenie , "Enlightenment"; pt, Iluminismo, "Enlightenment"; es, link=n ...
ideals of the French Revolution. When a chair of mathematics became vacant at the Collège de France in 1843, Cauchy applied for it, but received just three of 45 votes. The year 1848 was the year of revolution all over Europe; revolutions broke out in numerous countries, beginning in France. King Louis-Philippe, fearful of sharing the fate of Louis XVI, fled to England. The oath of allegiance was abolished, and the road to an academic appointment was finally clear for Cauchy. On March 1, 1849, he was reinstated at the Faculté de Sciences, as a professor of mathematical astronomy. After political turmoil all through 1848, France chose to become a Republic, under the Presidency of
Louis Napoleon Bonaparte
Louis Napoleon Bonaparte
, nephew of Napoleon Bonaparte, and son of Napoleon's brother, who had been installed as the first king of Holland. Soon (early 1852) the President made himself Emperor of France, and took the name
Napoleon III Napoleon III (Charles Louis Napoléon Bonaparte; 20 April 18089 January 1873) was the first President of France The president of France, officially the President of the French Republic (french: Président de la République française), is t ...

Napoleon III
. Not unexpectedly, the idea came up in bureaucratic circles that it would be useful to again require a loyalty oath from all state functionaries, including university professors. This time a cabinet minister was able to convince the Emperor to exempt Cauchy from the oath. Cauchy remained a professor at the University until his death at the age of 67. He received the
Last Rites #REDIRECT Last rites The last rites, in Catholicism, are the last prayers and ministrations given to an individual of the faith, when possible, shortly before death. They may be administered to those Death row, awaiting execution, mortally inju ...

Last Rites
and died of a bronchial condition at 4 a.m. on 23 May 1857. His name is one of the 72 names inscribed on the Eiffel Tower.


Work


Early work

The genius of Cauchy was illustrated in his simple solution of the
problem of Apollonius In Euclidean geometry, Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his wo ...
—describing a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
touching three given circles—which he discovered in 1805, his generalization of
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...
on
polyhedra In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polyhedra
in 1811, and in several other elegant problems. More important is his memoir on
wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

wave
propagation, which obtained the Grand Prix of the French Academy of Sciences in 1816. Cauchy's writings covered notable topics including: the theory of series, where he developed the notion of
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
and discovered many of the basic formulas for
q-series In mathematics, in the area of combinatorics, a ''q''-Pochhammer symbol, also called a ''q''-shifted factorial, is a q-analog, ''q''-analog of the Pochhammer symbol. It is defined as :(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^) w ...
. In the theory of numbers and complex quantities, he was the first to define complex numbers as pairs of real numbers. He also wrote on the theory of groups and substitutions, the theory of functions, differential equations and determinants.


Wave theory, mechanics, elasticity

In the theory of light he worked on Fresnel's wave theory and on the
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variation ...
and
polarization Polarization or polarisation may refer to: In the physical sciences *Polarization (waves), the ability of waves to oscillate in more than one direction, in particular polarization of light, responsible for example for the glare-reducing effect of ...
of light. He also contributed research in
mechanics Mechanics (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

mechanics
, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced a 3 × 3 symmetric
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
of numbers that is now known as the
Cauchy stress tensor In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis C ...
. In
elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the clu ...
, he originated the theory of stress, and his results are nearly as valuable as those of Siméon Poisson.


Number theory

Other significant contributions include being the first to prove the
Fermat polygonal number theorem In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the su ...
.


Complex functions

Cauchy is most famous for his single-handed development of
complex function theory of the function . Hue represents the argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also relat ...
. The first pivotal theorem proved by Cauchy, now known as ''
Cauchy's integral theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'', was the following: : \oint_C f(z)dz = 0, where ''f''(''z'') is a complex-valued function
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
on and within the non-self-intersecting closed curve ''C'' (contour) lying in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. The ''contour integral'' is taken along the contour ''C''. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form the theorem was given in 1825. The 1825 paper is seen by many as Cauchy's most important contribution to mathematics. In 1826 Cauchy gave a formal definition of a residue of a function. This concept concerns functions that have
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
s—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function ''f''(''z'') can be expanded in the
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

neighborhood
of a singularity ''a'' as : f(z) = \phi(z) + \frac + \frac + \cdots + \frac,\quad B_i, z,a \in \mathbb, where φ(''z'') is analytic (i.e., well-behaved without singularities), then ''f'' is said to have a pole of order ''n'' in the point ''a''. If ''n'' = 1, the pole is called simple. The coefficient ''B''1 is called by Cauchy the residue of function ''f'' at ''a''. If ''f'' is non-singular at ''a'' then the residue of ''f'' is zero at ''a''. Clearly the residue is in the case of a simple pole equal to, : \underset f(z) = \lim_ (z-a) f(z), where we replaced ''B''1 by the modern notation of the residue. In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first he proposed the formula now known as
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary ...
, : f(a) = \frac \oint_C \frac dz, where ''f''(''z'') is analytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at ''z'' = ''a''. In the second paperCauchy, ''Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu'offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes'' Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831. he presented the
residue theorem In complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
, : \frac \oint_C f(z) dz = \sum_^n \underset f(z), where the sum is over all the ''n'' poles of ''f''(''z'') on and within the contour ''C''. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with
Pierre Alphonse Laurent Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series (Holomorphic functions are analytic, analytic). In mathematics, the Laurent se ...
being the first mathematician, besides Cauchy, making a substantial contribution (his
Laurent series (Holomorphic functions are analytic, analytic). In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express compl ...

Laurent series
published in 1843).


Cours d'Analyse

In his book ''Cours d'Analyse'' Cauchy stressed the importance of rigor in analysis. ''Rigor'' in this case meant the rejection of the principle of ''
Generality of algebraIn the history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of ...
'' (of earlier authors such as Euler and Lagrange) and its replacement by geometry and
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe". The book is frequently noted as being the first place that inequalities, and \delta-\epsilon arguments were introduced into Calculus. Here Cauchy defined continuity as follows: ''The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.'' M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement . Gilain notes that when the portion of the curriculum devoted to ''Analyse Algébrique'' was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus . Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853. Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing they lead from everything from the usual "epsilontic" definitions or to the notions of
non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard ...
. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.


Taylor's theorem

He was the first to prove
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
rigorously, establishing his well-known form of the remainder. He wrote a textbook (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

limit
in the form that is still taught. Also Cauchy's well-known test for
absolute convergence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
stems from this book:
Cauchy condensation test In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by
Abel Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis The Book of Genesis,, "''Bərēšīṯ''", "In hebeginning" the first book of the Hebrew Bible The Hebrew Bible or Tanakh (; Hebrew: , or ), is the ...

Abel
, later fixed by the introduction of the notion of
uniform continuity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
.


Argument principle, stability

In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "
Argument Principle frame, The simple contour ''C'' (black), the zeros of ''f'' (blue) and the poles of ''f'' (red). Here we have \frac\oint_ \, dz=4-5.\, In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the n ...
" in many modern textbooks on complex analysis. In modern control theory textbooks, the Cauchy argument principle is quite frequently used to derive the Nyquist stability criterion, which can be used to predict the stability of negative feedback amplifier and negative feedback control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.


Published works

Cauchy was very productive, in number of papers second only to Leonhard Euler. It took almost a century to collect all his writings into 27 large volumes: * (Paris : Gauthier-Villars et fils, 1882–1974) * His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises: * * ''Le Calcul infinitésimal'' (1823) * ''Leçons sur les applications de calcul infinitésimal''; ''La géométrie'' (1826–1828) His other works include: * * * * * *
Exercices d'analyse et de physique mathematique (Volume 1)
' *
Exercices d'analyse et de physique mathematique (Volume 2)
' *
Exercices d'analyse et de physique mathematique (Volume 3)
' *
Exercices d'analyse et de physique mathematique (Volume 4)
' (Paris: Bachelier, 1840–1847) *
Analyse algèbrique
' (Imprimerie Royale, 1821) *
Nouveaux exercices de mathématiques
' (Paris : Gauthier-Villars, 1895) * ''Courses of mechanics'' (for the École Polytechnique) * ''Higher algebra'' (for the ) * ''Mathematical physics'' (for the Collège de France). *
Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis
' CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843) credited as originating the operational calculus.


Politics and religious beliefs

Augustin-Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to
Arcueil Arcueil () is a Communes of France, commune in the Val-de-Marne Departments of France, department in the southern suburbs of Paris, France. It is located from the Kilometre Zero, center of Paris. Name The name Arcueil was recorded for the fi ...
during the
French Revolution The French Revolution ( ) was a period of radical political and societal change in France France (), officially the French Republic (french: link=no, République française), is a spanning and in the and the , and s. Its ...

French Revolution
. Their life there during that time was apparently hard; Augustin-Louis's father, Louis François, spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis François to his mother in Rouen says: In any event, he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X. He was an equally staunch Catholic and a member of the Society of Saint Vincent de Paul. He also had links to the Society of Jesus and defended them at the Academy when it was politically unwise to do so. His zeal for his faith may have led to his caring for Charles Hermite during his illness and leading Hermite to become a faithful Catholic. It also inspired Cauchy to plead on behalf of the Irish during the Great Famine (Ireland), Great Famine of Ireland. His royalism and religious zeal also made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. Niels Henrik Abel called him a "bigoted Catholic" and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when Guglielmo Libri Carucci dalla Sommaja was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by Joseph Liouville rather than Cauchy, which caused a rift between Liouville and Cauchy. Another dispute with political overtones concerned Jean-Marie Duhamel, Jean-Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by Jean-Victor Poncelet, to be wrong.


See also

* List of topics named after Augustin-Louis Cauchy * Cauchy–Binet formula * Cauchy boundary condition * Cauchy's convergence test * Cauchy (crater) * Cauchy determinant * Cauchy distribution * Cauchy's equation * Cauchy–Euler equation * Cauchy's functional equation * Cauchy horizon * Cauchy formula for repeated integration * Cauchy–Frobenius lemma * Cauchy–Hadamard theorem * Cauchy–Kovalevskaya theorem * Cauchy momentum equation * Cauchy–Peano theorem * Cauchy principal value * Cauchy problem * Cauchy product * Cauchy's radical test * Cauchy–Rassias stability * Cauchy–Riemann equations * Cauchy–Schwarz inequality * Cauchy sequence * Cauchy surface * Cauchy's theorem (geometry) * Cauchy's theorem (group theory) * Maclaurin–Cauchy test


Notes


References

* * * * * * * * * * *


Further reading

* * * Boyer, C.: The concepts of the calculus. Hafner Publishing Company, 1949. * Benis-Sinaceur Hourya. Cauchy et Bolzano. In: Revue d'histoire des sciences. 1973, Tome 26 n°2. pp. 97–112. * . * * * *


External links

* *
Augustin-Louis Cauchy – Œuvres complètes
(in 2 series) Gallica-Math *
Augustin-Louis Cauchy – Cauchy's Life
by Robin Hartshorne {{DEFAULTSORT:Cauchy, Augustin Louis Augustin-Louis Cauchy, 1789 births 1857 deaths 19th-century French mathematicians Corps des ponts École des Ponts ParisTech alumni École Polytechnique alumni Fellows of the American Academy of Arts and Sciences Foreign Members of the Royal Society French Roman Catholics Geometers History of calculus Mathematical analysts Linear algebraists Members of the French Academy of Sciences Members of the Royal Swedish Academy of Sciences Recipients of the Pour le Mérite (civil class) Textbook writers University of Turin faculty